Integrand size = 24, antiderivative size = 1248 \[ \int \frac {1}{x^2 \left (a+b x^4\right )^2 \sqrt {c+d x^4}} \, dx =\text {Too large to display} \] Output:
-1/4*(-4*a*d+5*b*c)*(d*x^4+c)^(1/2)/a^2/c/(-a*d+b*c)/x+1/4*d^(1/2)*(-4*a*d +5*b*c)*x*(d*x^4+c)^(1/2)/a^2/c/(-a*d+b*c)/(c^(1/2)+d^(1/2)*x^2)+1/4*b*(d* x^4+c)^(1/2)/a/(-a*d+b*c)/x/(b*x^4+a)-1/16*b^(3/4)*(-7*a*d+5*b*c)*arctan(( -a*d+b*c)^(1/2)*x/(-a)^(1/4)/b^(1/4)/(d*x^4+c)^(1/2))/(-a)^(9/4)/(-a*d+b*c )^(3/2)+1/16*b^(3/4)*(-7*a*d+5*b*c)*arctanh((-a*d+b*c)^(1/2)*x/(-a)^(1/4)/ b^(1/4)/(d*x^4+c)^(1/2))/(-a)^(9/4)/(-a*d+b*c)^(3/2)-1/4*d^(1/4)*(-4*a*d+5 *b*c)*(c^(1/2)+d^(1/2)*x^2)*((d*x^4+c)/(c^(1/2)+d^(1/2)*x^2)^2)^(1/2)*Elli pticE(sin(2*arctan(d^(1/4)*x/c^(1/4))),1/2*2^(1/2))/a^2/c^(3/4)/(-a*d+b*c) /(d*x^4+c)^(1/2)+1/16*b^(1/2)*d^(1/4)*(-7*a*d+5*b*c)*(c^(1/2)+d^(1/2)*x^2) *((d*x^4+c)/(c^(1/2)+d^(1/2)*x^2)^2)^(1/2)*InverseJacobiAM(2*arctan(d^(1/4 )*x/c^(1/4)),1/2*2^(1/2))/a^2/c^(1/4)/(b^(1/2)*c^(1/2)-(-a)^(1/2)*d^(1/2)) /(-a*d+b*c)/(d*x^4+c)^(1/2)+1/16*b^(1/2)*d^(1/4)*(-7*a*d+5*b*c)*(c^(1/2)+d ^(1/2)*x^2)*((d*x^4+c)/(c^(1/2)+d^(1/2)*x^2)^2)^(1/2)*InverseJacobiAM(2*ar ctan(d^(1/4)*x/c^(1/4)),1/2*2^(1/2))/a^2/c^(1/4)/(b^(1/2)*c^(1/2)+(-a)^(1/ 2)*d^(1/2))/(-a*d+b*c)/(d*x^4+c)^(1/2)+1/8*d^(1/4)*(-4*a*d+5*b*c)*(c^(1/2) +d^(1/2)*x^2)*((d*x^4+c)/(c^(1/2)+d^(1/2)*x^2)^2)^(1/2)*InverseJacobiAM(2* arctan(d^(1/4)*x/c^(1/4)),1/2*2^(1/2))/a^2/c^(3/4)/(-a*d+b*c)/(d*x^4+c)^(1 /2)-1/32*b^(1/2)*(b^(1/2)*c^(1/2)+(-a)^(1/2)*d^(1/2))*(-7*a*d+5*b*c)*(c^(1 /2)+d^(1/2)*x^2)*((d*x^4+c)/(c^(1/2)+d^(1/2)*x^2)^2)^(1/2)*EllipticPi(sin( 2*arctan(d^(1/4)*x/c^(1/4))),-1/4*(b^(1/2)*c^(1/2)-(-a)^(1/2)*d^(1/2))^...
Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.
Time = 10.21 (sec) , antiderivative size = 226, normalized size of antiderivative = 0.18 \[ \int \frac {1}{x^2 \left (a+b x^4\right )^2 \sqrt {c+d x^4}} \, dx=\frac {21 a \left (c+d x^4\right ) \left (4 a^2 d-5 b^2 c x^4-4 a b \left (c-d x^4\right )\right )-7 \left (5 b^2 c^2-12 a b c d+4 a^2 d^2\right ) x^4 \left (a+b x^4\right ) \sqrt {1+\frac {d x^4}{c}} \operatorname {AppellF1}\left (\frac {3}{4},\frac {1}{2},1,\frac {7}{4},-\frac {d x^4}{c},-\frac {b x^4}{a}\right )+3 b d (5 b c-4 a d) x^8 \left (a+b x^4\right ) \sqrt {1+\frac {d x^4}{c}} \operatorname {AppellF1}\left (\frac {7}{4},\frac {1}{2},1,\frac {11}{4},-\frac {d x^4}{c},-\frac {b x^4}{a}\right )}{84 a^3 c (b c-a d) x \left (a+b x^4\right ) \sqrt {c+d x^4}} \] Input:
Integrate[1/(x^2*(a + b*x^4)^2*Sqrt[c + d*x^4]),x]
Output:
(21*a*(c + d*x^4)*(4*a^2*d - 5*b^2*c*x^4 - 4*a*b*(c - d*x^4)) - 7*(5*b^2*c ^2 - 12*a*b*c*d + 4*a^2*d^2)*x^4*(a + b*x^4)*Sqrt[1 + (d*x^4)/c]*AppellF1[ 3/4, 1/2, 1, 7/4, -((d*x^4)/c), -((b*x^4)/a)] + 3*b*d*(5*b*c - 4*a*d)*x^8* (a + b*x^4)*Sqrt[1 + (d*x^4)/c]*AppellF1[7/4, 1/2, 1, 11/4, -((d*x^4)/c), -((b*x^4)/a)])/(84*a^3*c*(b*c - a*d)*x*(a + b*x^4)*Sqrt[c + d*x^4])
Time = 2.43 (sec) , antiderivative size = 1148, normalized size of antiderivative = 0.92, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {972, 25, 1053, 1054, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {1}{x^2 \left (a+b x^4\right )^2 \sqrt {c+d x^4}} \, dx\) |
| \(\Big \downarrow \) 972 |
| \(\displaystyle \frac {b \sqrt {c+d x^4}}{4 a x \left (a+b x^4\right ) (b c-a d)}-\frac {\int -\frac {3 b d x^4+5 b c-4 a d}{x^2 \left (b x^4+a\right ) \sqrt {d x^4+c}}dx}{4 a (b c-a d)}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int \frac {3 b d x^4+5 b c-4 a d}{x^2 \left (b x^4+a\right ) \sqrt {d x^4+c}}dx}{4 a (b c-a d)}+\frac {b \sqrt {c+d x^4}}{4 a x \left (a+b x^4\right ) (b c-a d)}\) |
| \(\Big \downarrow \) 1053 |
| \(\displaystyle \frac {-\frac {\int \frac {x^2 \left ((b c-2 a d) (5 b c-2 a d)-b d (5 b c-4 a d) x^4\right )}{\left (b x^4+a\right ) \sqrt {d x^4+c}}dx}{a c}-\frac {\sqrt {c+d x^4} (5 b c-4 a d)}{a c x}}{4 a (b c-a d)}+\frac {b \sqrt {c+d x^4}}{4 a x \left (a+b x^4\right ) (b c-a d)}\) |
| \(\Big \downarrow \) 1054 |
| \(\displaystyle \frac {-\frac {\int \left (\frac {\left (5 b^2 c^2-7 a b c d\right ) x^2}{\left (b x^4+a\right ) \sqrt {d x^4+c}}-\frac {d (5 b c-4 a d) x^2}{\sqrt {d x^4+c}}\right )dx}{a c}-\frac {\sqrt {c+d x^4} (5 b c-4 a d)}{a c x}}{4 a (b c-a d)}+\frac {b \sqrt {c+d x^4}}{4 a x \left (a+b x^4\right ) (b c-a d)}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\sqrt {d x^4+c} b}{4 a (b c-a d) x \left (b x^4+a\right )}+\frac {-\frac {\sqrt {d x^4+c} (5 b c-4 a d)}{a c x}-\frac {-\frac {\sqrt {b} c^{3/4} (5 b c-7 a d) \left (\sqrt {d} x^2+\sqrt {c}\right ) \sqrt {\frac {d x^4+c}{\left (\sqrt {d} x^2+\sqrt {c}\right )^2}} \operatorname {EllipticPi}\left (\frac {\left (\sqrt {b} \sqrt {c}+\sqrt {-a} \sqrt {d}\right )^2}{4 \sqrt {-a} \sqrt {b} \sqrt {c} \sqrt {d}},2 \arctan \left (\frac {\sqrt [4]{d} x}{\sqrt [4]{c}}\right ),\frac {1}{2}\right ) \left (\sqrt {b} \sqrt {c}-\sqrt {-a} \sqrt {d}\right )^2}{8 \sqrt {-a} \sqrt [4]{d} (b c+a d) \sqrt {d x^4+c}}+\frac {b^{3/4} c (5 b c-7 a d) \arctan \left (\frac {\sqrt {b c-a d} x}{\sqrt [4]{-a} \sqrt [4]{b} \sqrt {d x^4+c}}\right )}{4 \sqrt [4]{-a} \sqrt {b c-a d}}-\frac {b^{3/4} c (5 b c-7 a d) \text {arctanh}\left (\frac {\sqrt {b c-a d} x}{\sqrt [4]{-a} \sqrt [4]{b} \sqrt {d x^4+c}}\right )}{4 \sqrt [4]{-a} \sqrt {b c-a d}}+\frac {\sqrt [4]{c} \sqrt [4]{d} (5 b c-4 a d) \left (\sqrt {d} x^2+\sqrt {c}\right ) \sqrt {\frac {d x^4+c}{\left (\sqrt {d} x^2+\sqrt {c}\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{d} x}{\sqrt [4]{c}}\right )|\frac {1}{2}\right )}{\sqrt {d x^4+c}}-\frac {\sqrt [4]{c} \sqrt [4]{d} (5 b c-4 a d) \left (\sqrt {d} x^2+\sqrt {c}\right ) \sqrt {\frac {d x^4+c}{\left (\sqrt {d} x^2+\sqrt {c}\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{d} x}{\sqrt [4]{c}}\right ),\frac {1}{2}\right )}{2 \sqrt {d x^4+c}}-\frac {b c^{3/4} \left (\sqrt {c}-\frac {\sqrt {-a} \sqrt {d}}{\sqrt {b}}\right ) \sqrt [4]{d} (5 b c-7 a d) \left (\sqrt {d} x^2+\sqrt {c}\right ) \sqrt {\frac {d x^4+c}{\left (\sqrt {d} x^2+\sqrt {c}\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{d} x}{\sqrt [4]{c}}\right ),\frac {1}{2}\right )}{4 (b c+a d) \sqrt {d x^4+c}}-\frac {b c^{3/4} \left (\sqrt {c}+\frac {\sqrt {-a} \sqrt {d}}{\sqrt {b}}\right ) \sqrt [4]{d} (5 b c-7 a d) \left (\sqrt {d} x^2+\sqrt {c}\right ) \sqrt {\frac {d x^4+c}{\left (\sqrt {d} x^2+\sqrt {c}\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{d} x}{\sqrt [4]{c}}\right ),\frac {1}{2}\right )}{4 (b c+a d) \sqrt {d x^4+c}}+\frac {\sqrt {b} c^{3/4} \left (\sqrt {b} \sqrt {c}+\sqrt {-a} \sqrt {d}\right )^2 (5 b c-7 a d) \left (\sqrt {d} x^2+\sqrt {c}\right ) \sqrt {\frac {d x^4+c}{\left (\sqrt {d} x^2+\sqrt {c}\right )^2}} \operatorname {EllipticPi}\left (-\frac {\sqrt {c} \left (\sqrt {b}-\frac {\sqrt {-a} \sqrt {d}}{\sqrt {c}}\right )^2}{4 \sqrt {-a} \sqrt {b} \sqrt {d}},2 \arctan \left (\frac {\sqrt [4]{d} x}{\sqrt [4]{c}}\right ),\frac {1}{2}\right )}{8 \sqrt {-a} \sqrt [4]{d} (b c+a d) \sqrt {d x^4+c}}-\frac {\sqrt {d} (5 b c-4 a d) x \sqrt {d x^4+c}}{\sqrt {d} x^2+\sqrt {c}}}{a c}}{4 a (b c-a d)}\) |
Input:
Int[1/(x^2*(a + b*x^4)^2*Sqrt[c + d*x^4]),x]
Output:
(b*Sqrt[c + d*x^4])/(4*a*(b*c - a*d)*x*(a + b*x^4)) + (-(((5*b*c - 4*a*d)* Sqrt[c + d*x^4])/(a*c*x)) - (-((Sqrt[d]*(5*b*c - 4*a*d)*x*Sqrt[c + d*x^4]) /(Sqrt[c] + Sqrt[d]*x^2)) + (b^(3/4)*c*(5*b*c - 7*a*d)*ArcTan[(Sqrt[b*c - a*d]*x)/((-a)^(1/4)*b^(1/4)*Sqrt[c + d*x^4])])/(4*(-a)^(1/4)*Sqrt[b*c - a* d]) - (b^(3/4)*c*(5*b*c - 7*a*d)*ArcTanh[(Sqrt[b*c - a*d]*x)/((-a)^(1/4)*b ^(1/4)*Sqrt[c + d*x^4])])/(4*(-a)^(1/4)*Sqrt[b*c - a*d]) + (c^(1/4)*d^(1/4 )*(5*b*c - 4*a*d)*(Sqrt[c] + Sqrt[d]*x^2)*Sqrt[(c + d*x^4)/(Sqrt[c] + Sqrt [d]*x^2)^2]*EllipticE[2*ArcTan[(d^(1/4)*x)/c^(1/4)], 1/2])/Sqrt[c + d*x^4] - (c^(1/4)*d^(1/4)*(5*b*c - 4*a*d)*(Sqrt[c] + Sqrt[d]*x^2)*Sqrt[(c + d*x^ 4)/(Sqrt[c] + Sqrt[d]*x^2)^2]*EllipticF[2*ArcTan[(d^(1/4)*x)/c^(1/4)], 1/2 ])/(2*Sqrt[c + d*x^4]) - (b*c^(3/4)*(Sqrt[c] - (Sqrt[-a]*Sqrt[d])/Sqrt[b]) *d^(1/4)*(5*b*c - 7*a*d)*(Sqrt[c] + Sqrt[d]*x^2)*Sqrt[(c + d*x^4)/(Sqrt[c] + Sqrt[d]*x^2)^2]*EllipticF[2*ArcTan[(d^(1/4)*x)/c^(1/4)], 1/2])/(4*(b*c + a*d)*Sqrt[c + d*x^4]) - (b*c^(3/4)*(Sqrt[c] + (Sqrt[-a]*Sqrt[d])/Sqrt[b] )*d^(1/4)*(5*b*c - 7*a*d)*(Sqrt[c] + Sqrt[d]*x^2)*Sqrt[(c + d*x^4)/(Sqrt[c ] + Sqrt[d]*x^2)^2]*EllipticF[2*ArcTan[(d^(1/4)*x)/c^(1/4)], 1/2])/(4*(b*c + a*d)*Sqrt[c + d*x^4]) - (Sqrt[b]*c^(3/4)*(Sqrt[b]*Sqrt[c] - Sqrt[-a]*Sq rt[d])^2*(5*b*c - 7*a*d)*(Sqrt[c] + Sqrt[d]*x^2)*Sqrt[(c + d*x^4)/(Sqrt[c] + Sqrt[d]*x^2)^2]*EllipticPi[(Sqrt[b]*Sqrt[c] + Sqrt[-a]*Sqrt[d])^2/(4*Sq rt[-a]*Sqrt[b]*Sqrt[c]*Sqrt[d]), 2*ArcTan[(d^(1/4)*x)/c^(1/4)], 1/2])/(...
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_ ))^(q_), x_Symbol] :> Simp[(-b)*(e*x)^(m + 1)*(a + b*x^n)^(p + 1)*((c + d*x ^n)^(q + 1)/(a*e*n*(b*c - a*d)*(p + 1))), x] + Simp[1/(a*n*(b*c - a*d)*(p + 1)) Int[(e*x)^m*(a + b*x^n)^(p + 1)*(c + d*x^n)^q*Simp[c*b*(m + 1) + n*( b*c - a*d)*(p + 1) + d*b*(m + n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{ a, b, c, d, e, m, q}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && LtQ[p, -1] & & IntBinomialQ[a, b, c, d, e, m, n, p, q, x]
Int[((g_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_ ))^(q_.)*((e_) + (f_.)*(x_)^(n_)), x_Symbol] :> Simp[e*(g*x)^(m + 1)*(a + b *x^n)^(p + 1)*((c + d*x^n)^(q + 1)/(a*c*g*(m + 1))), x] + Simp[1/(a*c*g^n*( m + 1)) Int[(g*x)^(m + n)*(a + b*x^n)^p*(c + d*x^n)^q*Simp[a*f*c*(m + 1) - e*(b*c + a*d)*(m + n + 1) - e*n*(b*c*p + a*d*q) - b*e*d*(m + n*(p + q + 2 ) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p, q}, x] && IGtQ[n, 0] && LtQ[m, -1]
Int[(((g_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((e_) + (f_.)*(x_)^(n _)))/((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Int[ExpandIntegrand[(g*x)^m*(a + b*x^n)^p*((e + f*x^n)/(c + d*x^n)), x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && IGtQ[n, 0]
Result contains complex when optimal does not.
Time = 8.73 (sec) , antiderivative size = 392, normalized size of antiderivative = 0.31
| method | result | size |
| elliptic | \(\frac {b^{2} x^{3} \sqrt {d \,x^{4}+c}}{4 a^{2} \left (a d -c b \right ) \left (b \,x^{4}+a \right )}-\frac {\sqrt {d \,x^{4}+c}}{c \,a^{2} x}+\frac {i \left (-\frac {b d}{4 a^{2} \left (a d -c b \right )}+\frac {d}{c \,a^{2}}\right ) \sqrt {c}\, \sqrt {1-\frac {i \sqrt {d}\, x^{2}}{\sqrt {c}}}\, \sqrt {1+\frac {i \sqrt {d}\, x^{2}}{\sqrt {c}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {d}}{\sqrt {c}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {i \sqrt {d}}{\sqrt {c}}}, i\right )\right )}{\sqrt {\frac {i \sqrt {d}}{\sqrt {c}}}\, \sqrt {d \,x^{4}+c}\, \sqrt {d}}-\frac {\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (\textit {\_Z}^{4} b +a \right )}{\sum }\frac {\left (7 a d -5 c b \right ) \left (-\frac {\operatorname {arctanh}\left (\frac {2 d \,x^{2} \underline {\hspace {1.25 ex}}\alpha ^{2}+2 c}{2 \sqrt {\frac {-a d +c b}{b}}\, \sqrt {d \,x^{4}+c}}\right )}{\sqrt {\frac {-a d +c b}{b}}}+\frac {2 \underline {\hspace {1.25 ex}}\alpha ^{3} b \sqrt {1-\frac {i \sqrt {d}\, x^{2}}{\sqrt {c}}}\, \sqrt {1+\frac {i \sqrt {d}\, x^{2}}{\sqrt {c}}}\, \operatorname {EllipticPi}\left (x \sqrt {\frac {i \sqrt {d}}{\sqrt {c}}}, \frac {i \sqrt {c}\, \underline {\hspace {1.25 ex}}\alpha ^{2} b}{\sqrt {d}\, a}, \frac {\sqrt {-\frac {i \sqrt {d}}{\sqrt {c}}}}{\sqrt {\frac {i \sqrt {d}}{\sqrt {c}}}}\right )}{\sqrt {\frac {i \sqrt {d}}{\sqrt {c}}}\, a \sqrt {d \,x^{4}+c}}\right )}{\left (a d -c b \right ) \underline {\hspace {1.25 ex}}\alpha }}{32 a^{2}}\) | \(392\) |
| default | \(\frac {-\frac {\sqrt {d \,x^{4}+c}}{c x}+\frac {i \sqrt {d}\, \sqrt {1-\frac {i \sqrt {d}\, x^{2}}{\sqrt {c}}}\, \sqrt {1+\frac {i \sqrt {d}\, x^{2}}{\sqrt {c}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {d}}{\sqrt {c}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {i \sqrt {d}}{\sqrt {c}}}, i\right )\right )}{\sqrt {c}\, \sqrt {\frac {i \sqrt {d}}{\sqrt {c}}}\, \sqrt {d \,x^{4}+c}}}{a^{2}}-\frac {b \left (-\frac {b \,x^{3} \sqrt {d \,x^{4}+c}}{4 a \left (a d -c b \right ) \left (b \,x^{4}+a \right )}+\frac {i \sqrt {d}\, \sqrt {c}\, \sqrt {1-\frac {i \sqrt {d}\, x^{2}}{\sqrt {c}}}\, \sqrt {1+\frac {i \sqrt {d}\, x^{2}}{\sqrt {c}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {d}}{\sqrt {c}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {i \sqrt {d}}{\sqrt {c}}}, i\right )\right )}{4 \left (a d -c b \right ) a \sqrt {\frac {i \sqrt {d}}{\sqrt {c}}}\, \sqrt {d \,x^{4}+c}}-\frac {\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (\textit {\_Z}^{4} b +a \right )}{\sum }\frac {\left (-3 a d +c b \right ) \left (-\frac {\operatorname {arctanh}\left (\frac {2 d \,x^{2} \underline {\hspace {1.25 ex}}\alpha ^{2}+2 c}{2 \sqrt {\frac {-a d +c b}{b}}\, \sqrt {d \,x^{4}+c}}\right )}{\sqrt {\frac {-a d +c b}{b}}}+\frac {2 \underline {\hspace {1.25 ex}}\alpha ^{3} b \sqrt {1-\frac {i \sqrt {d}\, x^{2}}{\sqrt {c}}}\, \sqrt {1+\frac {i \sqrt {d}\, x^{2}}{\sqrt {c}}}\, \operatorname {EllipticPi}\left (x \sqrt {\frac {i \sqrt {d}}{\sqrt {c}}}, \frac {i \sqrt {c}\, \underline {\hspace {1.25 ex}}\alpha ^{2} b}{\sqrt {d}\, a}, \frac {\sqrt {-\frac {i \sqrt {d}}{\sqrt {c}}}}{\sqrt {\frac {i \sqrt {d}}{\sqrt {c}}}}\right )}{\sqrt {\frac {i \sqrt {d}}{\sqrt {c}}}\, a \sqrt {d \,x^{4}+c}}\right )}{\left (a d -c b \right ) \underline {\hspace {1.25 ex}}\alpha }}{32 b a}\right )}{a}-\frac {\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (\textit {\_Z}^{4} b +a \right )}{\sum }\frac {-\frac {\operatorname {arctanh}\left (\frac {2 d \,x^{2} \underline {\hspace {1.25 ex}}\alpha ^{2}+2 c}{2 \sqrt {\frac {-a d +c b}{b}}\, \sqrt {d \,x^{4}+c}}\right )}{\sqrt {\frac {-a d +c b}{b}}}+\frac {2 \underline {\hspace {1.25 ex}}\alpha ^{3} b \sqrt {1-\frac {i \sqrt {d}\, x^{2}}{\sqrt {c}}}\, \sqrt {1+\frac {i \sqrt {d}\, x^{2}}{\sqrt {c}}}\, \operatorname {EllipticPi}\left (x \sqrt {\frac {i \sqrt {d}}{\sqrt {c}}}, \frac {i \sqrt {c}\, \underline {\hspace {1.25 ex}}\alpha ^{2} b}{\sqrt {d}\, a}, \frac {\sqrt {-\frac {i \sqrt {d}}{\sqrt {c}}}}{\sqrt {\frac {i \sqrt {d}}{\sqrt {c}}}}\right )}{\sqrt {\frac {i \sqrt {d}}{\sqrt {c}}}\, a \sqrt {d \,x^{4}+c}}}{\underline {\hspace {1.25 ex}}\alpha }}{8 a^{2}}\) | \(674\) |
| risch | \(-\frac {\sqrt {d \,x^{4}+c}}{c \,a^{2} x}+\frac {\frac {i \sqrt {d}\, \sqrt {c}\, \sqrt {1-\frac {i \sqrt {d}\, x^{2}}{\sqrt {c}}}\, \sqrt {1+\frac {i \sqrt {d}\, x^{2}}{\sqrt {c}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {d}}{\sqrt {c}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {i \sqrt {d}}{\sqrt {c}}}, i\right )\right )}{\sqrt {\frac {i \sqrt {d}}{\sqrt {c}}}\, \sqrt {d \,x^{4}+c}}-\frac {c \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (\textit {\_Z}^{4} b +a \right )}{\sum }\frac {-\frac {\operatorname {arctanh}\left (\frac {2 d \,x^{2} \underline {\hspace {1.25 ex}}\alpha ^{2}+2 c}{2 \sqrt {\frac {-a d +c b}{b}}\, \sqrt {d \,x^{4}+c}}\right )}{\sqrt {\frac {-a d +c b}{b}}}+\frac {2 \underline {\hspace {1.25 ex}}\alpha ^{3} b \sqrt {1-\frac {i \sqrt {d}\, x^{2}}{\sqrt {c}}}\, \sqrt {1+\frac {i \sqrt {d}\, x^{2}}{\sqrt {c}}}\, \operatorname {EllipticPi}\left (x \sqrt {\frac {i \sqrt {d}}{\sqrt {c}}}, \frac {i \sqrt {c}\, \underline {\hspace {1.25 ex}}\alpha ^{2} b}{\sqrt {d}\, a}, \frac {\sqrt {-\frac {i \sqrt {d}}{\sqrt {c}}}}{\sqrt {\frac {i \sqrt {d}}{\sqrt {c}}}}\right )}{\sqrt {\frac {i \sqrt {d}}{\sqrt {c}}}\, a \sqrt {d \,x^{4}+c}}}{\underline {\hspace {1.25 ex}}\alpha }\right )}{8}-a b c \left (-\frac {b \,x^{3} \sqrt {d \,x^{4}+c}}{4 a \left (a d -c b \right ) \left (b \,x^{4}+a \right )}+\frac {i \sqrt {d}\, \sqrt {c}\, \sqrt {1-\frac {i \sqrt {d}\, x^{2}}{\sqrt {c}}}\, \sqrt {1+\frac {i \sqrt {d}\, x^{2}}{\sqrt {c}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {d}}{\sqrt {c}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {i \sqrt {d}}{\sqrt {c}}}, i\right )\right )}{4 \left (a d -c b \right ) a \sqrt {\frac {i \sqrt {d}}{\sqrt {c}}}\, \sqrt {d \,x^{4}+c}}-\frac {\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (\textit {\_Z}^{4} b +a \right )}{\sum }\frac {\left (-3 a d +c b \right ) \left (-\frac {\operatorname {arctanh}\left (\frac {2 d \,x^{2} \underline {\hspace {1.25 ex}}\alpha ^{2}+2 c}{2 \sqrt {\frac {-a d +c b}{b}}\, \sqrt {d \,x^{4}+c}}\right )}{\sqrt {\frac {-a d +c b}{b}}}+\frac {2 \underline {\hspace {1.25 ex}}\alpha ^{3} b \sqrt {1-\frac {i \sqrt {d}\, x^{2}}{\sqrt {c}}}\, \sqrt {1+\frac {i \sqrt {d}\, x^{2}}{\sqrt {c}}}\, \operatorname {EllipticPi}\left (x \sqrt {\frac {i \sqrt {d}}{\sqrt {c}}}, \frac {i \sqrt {c}\, \underline {\hspace {1.25 ex}}\alpha ^{2} b}{\sqrt {d}\, a}, \frac {\sqrt {-\frac {i \sqrt {d}}{\sqrt {c}}}}{\sqrt {\frac {i \sqrt {d}}{\sqrt {c}}}}\right )}{\sqrt {\frac {i \sqrt {d}}{\sqrt {c}}}\, a \sqrt {d \,x^{4}+c}}\right )}{\left (a d -c b \right ) \underline {\hspace {1.25 ex}}\alpha }}{32 b a}\right )}{a^{2} c}\) | \(677\) |
Input:
int(1/x^2/(b*x^4+a)^2/(d*x^4+c)^(1/2),x,method=_RETURNVERBOSE)
Output:
1/4*b^2/a^2/(a*d-b*c)*x^3*(d*x^4+c)^(1/2)/(b*x^4+a)-1/c/a^2*(d*x^4+c)^(1/2 )/x+I*(-1/4*b*d/a^2/(a*d-b*c)+1/c*d/a^2)*c^(1/2)/(I/c^(1/2)*d^(1/2))^(1/2) *(1-I/c^(1/2)*d^(1/2)*x^2)^(1/2)*(1+I/c^(1/2)*d^(1/2)*x^2)^(1/2)/(d*x^4+c) ^(1/2)/d^(1/2)*(EllipticF(x*(I/c^(1/2)*d^(1/2))^(1/2),I)-EllipticE(x*(I/c^ (1/2)*d^(1/2))^(1/2),I))-1/32/a^2*sum((7*a*d-5*b*c)/(a*d-b*c)/_alpha*(-1/( (-a*d+b*c)/b)^(1/2)*arctanh(1/2*(2*_alpha^2*d*x^2+2*c)/((-a*d+b*c)/b)^(1/2 )/(d*x^4+c)^(1/2))+2/(I/c^(1/2)*d^(1/2))^(1/2)*_alpha^3*b/a*(1-I/c^(1/2)*d ^(1/2)*x^2)^(1/2)*(1+I/c^(1/2)*d^(1/2)*x^2)^(1/2)/(d*x^4+c)^(1/2)*Elliptic Pi(x*(I/c^(1/2)*d^(1/2))^(1/2),I*c^(1/2)/d^(1/2)*_alpha^2/a*b,(-I/c^(1/2)* d^(1/2))^(1/2)/(I/c^(1/2)*d^(1/2))^(1/2))),_alpha=RootOf(_Z^4*b+a))
Timed out. \[ \int \frac {1}{x^2 \left (a+b x^4\right )^2 \sqrt {c+d x^4}} \, dx=\text {Timed out} \] Input:
integrate(1/x^2/(b*x^4+a)^2/(d*x^4+c)^(1/2),x, algorithm="fricas")
Output:
Timed out
\[ \int \frac {1}{x^2 \left (a+b x^4\right )^2 \sqrt {c+d x^4}} \, dx=\int \frac {1}{x^{2} \left (a + b x^{4}\right )^{2} \sqrt {c + d x^{4}}}\, dx \] Input:
integrate(1/x**2/(b*x**4+a)**2/(d*x**4+c)**(1/2),x)
Output:
Integral(1/(x**2*(a + b*x**4)**2*sqrt(c + d*x**4)), x)
\[ \int \frac {1}{x^2 \left (a+b x^4\right )^2 \sqrt {c+d x^4}} \, dx=\int { \frac {1}{{\left (b x^{4} + a\right )}^{2} \sqrt {d x^{4} + c} x^{2}} \,d x } \] Input:
integrate(1/x^2/(b*x^4+a)^2/(d*x^4+c)^(1/2),x, algorithm="maxima")
Output:
integrate(1/((b*x^4 + a)^2*sqrt(d*x^4 + c)*x^2), x)
\[ \int \frac {1}{x^2 \left (a+b x^4\right )^2 \sqrt {c+d x^4}} \, dx=\int { \frac {1}{{\left (b x^{4} + a\right )}^{2} \sqrt {d x^{4} + c} x^{2}} \,d x } \] Input:
integrate(1/x^2/(b*x^4+a)^2/(d*x^4+c)^(1/2),x, algorithm="giac")
Output:
integrate(1/((b*x^4 + a)^2*sqrt(d*x^4 + c)*x^2), x)
Timed out. \[ \int \frac {1}{x^2 \left (a+b x^4\right )^2 \sqrt {c+d x^4}} \, dx=\int \frac {1}{x^2\,{\left (b\,x^4+a\right )}^2\,\sqrt {d\,x^4+c}} \,d x \] Input:
int(1/(x^2*(a + b*x^4)^2*(c + d*x^4)^(1/2)),x)
Output:
int(1/(x^2*(a + b*x^4)^2*(c + d*x^4)^(1/2)), x)
\[ \int \frac {1}{x^2 \left (a+b x^4\right )^2 \sqrt {c+d x^4}} \, dx=\frac {-\sqrt {d \,x^{4}+c}-3 \left (\int \frac {\sqrt {d \,x^{4}+c}\, x^{6}}{b^{2} d \,x^{12}+2 a b d \,x^{8}+b^{2} c \,x^{8}+a^{2} d \,x^{4}+2 a b c \,x^{4}+a^{2} c}d x \right ) a b d x -3 \left (\int \frac {\sqrt {d \,x^{4}+c}\, x^{6}}{b^{2} d \,x^{12}+2 a b d \,x^{8}+b^{2} c \,x^{8}+a^{2} d \,x^{4}+2 a b c \,x^{4}+a^{2} c}d x \right ) b^{2} d \,x^{5}+\left (\int \frac {\sqrt {d \,x^{4}+c}\, x^{2}}{b^{2} d \,x^{12}+2 a b d \,x^{8}+b^{2} c \,x^{8}+a^{2} d \,x^{4}+2 a b c \,x^{4}+a^{2} c}d x \right ) a^{2} d x -5 \left (\int \frac {\sqrt {d \,x^{4}+c}\, x^{2}}{b^{2} d \,x^{12}+2 a b d \,x^{8}+b^{2} c \,x^{8}+a^{2} d \,x^{4}+2 a b c \,x^{4}+a^{2} c}d x \right ) a b c x +\left (\int \frac {\sqrt {d \,x^{4}+c}\, x^{2}}{b^{2} d \,x^{12}+2 a b d \,x^{8}+b^{2} c \,x^{8}+a^{2} d \,x^{4}+2 a b c \,x^{4}+a^{2} c}d x \right ) a b d \,x^{5}-5 \left (\int \frac {\sqrt {d \,x^{4}+c}\, x^{2}}{b^{2} d \,x^{12}+2 a b d \,x^{8}+b^{2} c \,x^{8}+a^{2} d \,x^{4}+2 a b c \,x^{4}+a^{2} c}d x \right ) b^{2} c \,x^{5}}{a c x \left (b \,x^{4}+a \right )} \] Input:
int(1/x^2/(b*x^4+a)^2/(d*x^4+c)^(1/2),x)
Output:
( - sqrt(c + d*x**4) - 3*int((sqrt(c + d*x**4)*x**6)/(a**2*c + a**2*d*x**4 + 2*a*b*c*x**4 + 2*a*b*d*x**8 + b**2*c*x**8 + b**2*d*x**12),x)*a*b*d*x - 3*int((sqrt(c + d*x**4)*x**6)/(a**2*c + a**2*d*x**4 + 2*a*b*c*x**4 + 2*a*b *d*x**8 + b**2*c*x**8 + b**2*d*x**12),x)*b**2*d*x**5 + int((sqrt(c + d*x** 4)*x**2)/(a**2*c + a**2*d*x**4 + 2*a*b*c*x**4 + 2*a*b*d*x**8 + b**2*c*x**8 + b**2*d*x**12),x)*a**2*d*x - 5*int((sqrt(c + d*x**4)*x**2)/(a**2*c + a** 2*d*x**4 + 2*a*b*c*x**4 + 2*a*b*d*x**8 + b**2*c*x**8 + b**2*d*x**12),x)*a* b*c*x + int((sqrt(c + d*x**4)*x**2)/(a**2*c + a**2*d*x**4 + 2*a*b*c*x**4 + 2*a*b*d*x**8 + b**2*c*x**8 + b**2*d*x**12),x)*a*b*d*x**5 - 5*int((sqrt(c + d*x**4)*x**2)/(a**2*c + a**2*d*x**4 + 2*a*b*c*x**4 + 2*a*b*d*x**8 + b**2 *c*x**8 + b**2*d*x**12),x)*b**2*c*x**5)/(a*c*x*(a + b*x**4))